I'm returning to EVE Online after a fairly long break, and looking to get back into combat missions.

I've been trying to run Level 2 missions in my Dominix, with most of my DPS coming from drones. However, it seems that the types of drones I have can't contend with the quick ships of the enemies, and the gun turrets I have equipped just can't track the buggers.

I'd like to work my way up to doing Level 4's again; but gradually.

So, should I downgrade my ship? And what level of ship is most size-appropriate for level 2, 3, 4 missions?

The larger a turret, the slower it can turn to hit targets. A Dominix uses large turrets and these are basically useless against Frigate and Destroyer sized targets. Adding tracking computers will help a small bit, but stepping down to a Cruiser or Battlecruiser (and medium sized turrets) will improve tracking against Frigate sized targets. Alternatively, you may wish to fit a webifier to hold down a target while blasting it with your guns. As a comparison, here are some stats for 125mm, 250mm and 425mm railguns. In the "targetting" part of the "other attributes" section, look at "tracking speed": the small turret slews 3.6x the speed of the medium turret and the medium turret slews 2.4x faster than the large turret. A tracking computer with a tracking speed script will help significantly to improve the tracking speed.

The smaller ships are engaging in what is called "speed tanking" which means moving so fast your guns cannot keep up with the targets.

Generally, missiles are affected much less by the target's speed than turret weapons and this is part of why Drakes were the most popular missioning ship. The changes in Retribution 1.1 were an attempt to make the Drake less popular for missioning.

Mad Scientist's answer (frigates/destroyers, cruisers, battlecruisers, battleships) is the standard template, but it's worth noting that you can be much more effective with your Dominix in lower level missions if you mount medium or even small turrets. This allows you to easily take out any small stuff orbiting your ship while your drones tear up everything else. You'll save time this way since you won't have to send out and recall a wave of lights first. Note that I haven't tried this since the NPC AI change, so you may need to hit them with ECM first to make sure everything is targeting you rather than your drones.

Autocannons are probably your best bet due to good tracking and better damage projection over other short range turrets (because of falloff). You might also try rockets for the same reasons.